I was reading the book Elemental Methods in Ergodic Ramsey Theory. In the book they define the following: $$\Omega =\{1,\cdots, r\}^{\mathbb{N}}, \quad T:\Omega \to \Omega \quad \text{ defined by } T\gamma :=T\gamma(n)=\gamma(n+1) $$ $\Omega$ is cleary a metric space with metric $\rho(\gamma,\xi):=\frac{1}{min\{k: \gamma(k)\neq \xi(k)\}}$.
Also, it is clear to me that $T:\Omega\to \Omega$ is continuous. Let $\alpha\in \Omega$ If $X=\overline{\{T^{m}\alpha\mid m\in \mathbb{N}\}}$, then why $T\upharpoonright X$ is a continuous self-map of X? I was trying to prove that $T(X)\subseteq X$ proving that for all $r>0$ $$B(f(x),r)\cap \{T^{m}\alpha\mid m\in \mathbb{N}\}\neq \emptyset$$ but I can not conclude, I would appreciate some guidance for this exercise.
